Quantified Constraint Satisfaction and Small Commutative Conservative Operations

Abstract

The constraint satisfaction problem (CSP) is a broad framework capturing many combinatorial search problems. A natural and strict generalization of the CSP is the quantified constraint satisfaction problem (QCSP). The CSP involves deciding the truth of constraint networks where all variables are existentially quantified; the QCSP is defined similarly, but variables may be both existentially and universally quantified. While the CSP and QCSP are in their general formulation intractable, they can be parameterized by restricting the constraint language, that is, the types of constraints that are permitted in problem instances. Much attention has been directed towards classifying the complexity of all constraint languages in the case of the CSP. In this paper, we continue the recently initiated study of QCSP complexity by identifying a new family of tractable constraint languages, namely, constraint languages over domains of small size that are closed under a commutative conservative operation. This gives the first QCSP tractability result based on binary operations which may be non-associative. We also give a complete classification of maximal constraint languages over domains of size three.